Chapter 18, Problem 7PS

Summary Introduction

(a)

To calculate:

The geometric average time-weighted rates of return of portfolio

Introduction:

Annual geometric return:

It means a return on investment which shows that how much return investor has earned every year over period of investment.

Answer to Problem 7PS

  $\text{Annual geometric return is 3}\text{.57\%}$

Explanation of Solution

Given:

Price history of the stocks are given below:

Time	Price	Action
0	$\$90$	Buy 3 shares
1	$\$100$	Sell 1 share
2	$\$100$	Sell 1 share
3	$\$100$	Sell 1 share

Calculation of every year return:

  $\mathrm{Return}=\frac{\text{ Closing price}-\text{Opening price}}{\text{Opening price }}$

Time	Cash flow	Return
0	$3\times (-\$90)=-\$270$
1	$1\times \$100=\$100$	$\frac{100-90}{90}=11.11\%$
2	$1\times \$100=\$100$	$\frac{100-100}{100}=0\%$
3	$1\times \$100=\$100$	$\frac{100-100}{100}=0\%$

Calculation of annual geometric return:

  $\begin{array}{c}\text{Annual geometric return}={\left[\left(1+r_1\right)\times \left(1+r_2\right)\times \mathrm{...........}\times \left(1+r_n\right)\right]}^{\frac{1}{n}}-1\\ ={\left[\left(1+11.11\%\right)\times \left(1+0\%\right)\times \left(1+0\%\right)\right]}^{\frac{1}{3}}-1\\ ={\left(1.1111\right)}^{\frac{1}{3}}-1\\ =1.0357-1\\ =0.0357\text{ or 3}\text{.57\%}\end{array}$

Summary Introduction

(b)

To calculate:

Arithmetic average time-weighted rates of return of portfolio.

Introduction:

Arithmetic return:

An average return of the total years of investment is known as an arithmetic return.

Answer to Problem 7PS

  $Arithmeticaveragereturnis\text{ 3}\text{.7\%}$

Explanation of Solution

Given:

Price history of the stocks are given below:

Time	Price	Action
0	$\$90$	Buy 3 shares
1	$\$100$	Sell 1 share
2	$\$100$	Sell 1 share
3	$\$100$	Sell 1 share

Calculation of every year return:

  $\mathrm{Return}=\frac{\text{ Closing price}-\text{Opening price}}{\text{Opening price }}$

Time	Cash flow	Return
0	$3\times (-\$90)=-\$270$
1	$1\times \$100=\$100$	$\frac{100-90}{90}=11.11\%$
2	$1\times \$100=\$100$	$\frac{100-100}{100}=0\%$
3	$1\times \$100=\$100$	$\frac{100-100}{100}=0\%$

  $\begin{array}{c}Arithmeticaveragereturn=\frac{11.11+0+0}{3}\\ =3.7\%\end{array}$

Summary Introduction

(c)

To calculate:

The dollar-weighted average return of portfolio

Introduction:

Dollar-weighted return:

A return on portfolio or investment which also consider the timing of flows is known as dollar-weighted return.

Answer to Problem 7PS

Dollar-weighted return is equal to internal rate of return is $5.459\%$ .

Explanation of Solution

Calculation of cashflow:

Year	Cash flow	Explanation
0	$3\times (-\$90)=-\$270$	Three shares purchase at   $\$90\text{ per share}$
1	$1\times \$100=\$100$	one share sold at   $\$100\text{ per share}$
2	$1\times \$100=\$100$	one share sold at   $\$100\text{ per share}$
3	$1\times \$100=\$100$	one share sold at   $\$100\text{ per share}$

Calculation of IRR

  $\begin{array}{c}\text{NPV = Initial investment + }\frac{C_1}{{\left(1+r\right)}^1}+\frac{C_2}{{\left(1+r\right)}^2}+\mathrm{............}+\frac{C_n}{\begin{array}{l}{\left(1+r\right)}^n\\ \end{array}}\\ 0=-270+\frac{100}{{\left(1+r\right)}^1}+\frac{100}{{\left(1+r\right)}^2}+\frac{100}{{\left(1+r\right)}^3}\\ r=5.459\%\end{array}$

Thus, the dollar-weighted return is equal to internal rate of return which is $5.459\%$